Crude oil $($specific gravity $0\mathrm{.}86,$ kinematic viscosity $9\mathrm{.}0{10}^{-5}{m}^2{s}^{-1})$ is to be pumped from a barge to a large storage tank. The pipeline is horizontal and of diameter $250$ mm $,$ length $400$ m and roughness $0\mathrm{.}1$ mm $.$ It enters the tank $8$ m below the level of oil in the tank. When the control valve is fully open the static pressure at pump delivery is $3{10}^{5}Pa$ gauge. Ignore minor losses due to pipe fittings, entrance$/$exit losses etc.

Find:

$($a$)($using hydrostatics$)$ the gauge pressure where the pipe enters the tank;

$($b$)($from the pressures at the two ends$)$ the head loss along the pipeline;

$($c$)$ the volumetric flow rate in the pipeline.

If the pump delivery pressure remains the same but a valve reduces the flow by half, find:

$($d$)$ the head loss at the valve;

$($e$)$ the power loss at the valve.